Criteria for the stability of spatial extensions of fixed points and periodic orbits of differential equations in dimension 2

We consider partial differential equations that can be viewed as spatial extensions of two-dimensional differential equations with respect to a coupling matrix. We provide analytical and geometrical criteria governing the stability of spatially homogeneous stationary solutions and the phase stability of spatially homogeneous time periodic solutions. We distinguish cases where the differential equation has a conservative or a dissipative behavior. The geometrical criteria roughly speaking relate to the “sense of rotation” of, on one hand, the coupling matrix, and, on the other, the flow of the differential equation around the spatially homogeneous solution.